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alkaline-ml / statsmodels python
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# -*- coding: utf-8 -*-
"""Tests of GLSAR and diagnostics against Gretl
Created on Thu Feb 02 21:15:47 2012
Author: Josef Perktold
License: BSD-3
"""
import os
import numpy as np
from numpy.testing import (assert_almost_equal, assert_equal,
assert_allclose, assert_array_less)
from statsmodels.regression.linear_model import OLS, GLSAR
from statsmodels.tools.tools import add_constant
from statsmodels.datasets import macrodata
import statsmodels.stats.sandwich_covariance as sw
import statsmodels.stats.diagnostic as smsdia
import statsmodels.stats.outliers_influence as oi
def compare_ftest(contrast_res, other, decimal=(5,4)):
assert_almost_equal(contrast_res.fvalue, other[0], decimal=decimal[0])
assert_almost_equal(contrast_res.pvalue, other[1], decimal=decimal[1])
assert_equal(contrast_res.df_num, other[2])
assert_equal(contrast_res.df_denom, other[3])
assert_equal("f", other[4])
class TestGLSARGretl(object):
def test_all(self):
d = macrodata.load_pandas().data
#import datasetswsm.greene as g
#d = g.load('5-1')
#growth rates
gs_l_realinv = 400 * np.diff(np.log(d['realinv'].values))
gs_l_realgdp = 400 * np.diff(np.log(d['realgdp'].values))
#simple diff, not growthrate, I want heteroscedasticity later for testing
endogd = np.diff(d['realinv'])
exogd = add_constant(np.c_[np.diff(d['realgdp'].values), d['realint'][:-1].values])
endogg = gs_l_realinv
exogg = add_constant(np.c_[gs_l_realgdp, d['realint'][:-1].values])
res_ols = OLS(endogg, exogg).fit()
#print res_ols.params
mod_g1 = GLSAR(endogg, exogg, rho=-0.108136)
res_g1 = mod_g1.fit()
#print res_g1.params
mod_g2 = GLSAR(endogg, exogg, rho=-0.108136) #-0.1335859) from R
res_g2 = mod_g2.iterative_fit(maxiter=5)
#print res_g2.params
rho = -0.108136
# coefficient std. error t-ratio p-value 95% CONFIDENCE INTERVAL
partable = np.array([
[-9.50990, 0.990456, -9.602, 3.65e-018, -11.4631, -7.55670], # ***
[ 4.37040, 0.208146, 21.00, 2.93e-052, 3.95993, 4.78086], # ***
[-0.579253, 0.268009, -2.161, 0.0319, -1.10777, -0.0507346]]) # **
#Statistics based on the rho-differenced data:
result_gretl_g1 = dict(
endog_mean = ("Mean dependent var", 3.113973),
endog_std = ("S.D. dependent var", 18.67447),
ssr = ("Sum squared resid", 22530.90),
mse_resid_sqrt = ("S.E. of regression", 10.66735),
rsquared = ("R-squared", 0.676973),
rsquared_adj = ("Adjusted R-squared", 0.673710),
fvalue = ("F(2, 198)", 221.0475),
f_pvalue = ("P-value(F)", 3.56e-51),
resid_acf1 = ("rho", -0.003481),
dw = ("Durbin-Watson", 1.993858))
#fstatistic, p-value, df1, df2
reset_2_3 = [5.219019, 0.00619, 2, 197, "f"]
reset_2 = [7.268492, 0.00762, 1, 198, "f"]
reset_3 = [5.248951, 0.023, 1, 198, "f"]
#LM-statistic, p-value, df
arch_4 = [7.30776, 0.120491, 4, "chi2"]
#multicollinearity
vif = [1.002, 1.002]
cond_1norm = 6862.0664
determinant = 1.0296049e+009
reciprocal_condition_number = 0.013819244
#Chi-square(2): test-statistic, pvalue, df
normality = [20.2792, 3.94837e-005, 2]
#tests
res = res_g1 #with rho from Gretl
#basic
assert_almost_equal(res.params, partable[:,0], 4)
assert_almost_equal(res.bse, partable[:,1], 6)
assert_almost_equal(res.tvalues, partable[:,2], 2)
assert_almost_equal(res.ssr, result_gretl_g1['ssr'][1], decimal=2)
#assert_almost_equal(res.llf, result_gretl_g1['llf'][1], decimal=7) #not in gretl
#assert_almost_equal(res.rsquared, result_gretl_g1['rsquared'][1], decimal=7) #FAIL
#assert_almost_equal(res.rsquared_adj, result_gretl_g1['rsquared_adj'][1], decimal=7) #FAIL
assert_almost_equal(np.sqrt(res.mse_resid), result_gretl_g1['mse_resid_sqrt'][1], decimal=5)
assert_almost_equal(res.fvalue, result_gretl_g1['fvalue'][1], decimal=4)
assert_allclose(res.f_pvalue,
result_gretl_g1['f_pvalue'][1],
rtol=1e-2)
#assert_almost_equal(res.durbin_watson, result_gretl_g1['dw'][1], decimal=7) #TODO
#arch
#sm_arch = smsdia.acorr_lm(res.wresid**2, maxlag=4, autolag=None)
sm_arch = smsdia.het_arch(res.wresid, maxlag=4)
assert_almost_equal(sm_arch[0], arch_4[0], decimal=4)
assert_almost_equal(sm_arch[1], arch_4[1], decimal=6)
#tests
res = res_g2 #with estimated rho
#estimated lag coefficient
assert_almost_equal(res.model.rho, rho, decimal=3)
#basic
assert_almost_equal(res.params, partable[:,0], 4)
assert_almost_equal(res.bse, partable[:,1], 3)
assert_almost_equal(res.tvalues, partable[:,2], 2)
assert_almost_equal(res.ssr, result_gretl_g1['ssr'][1], decimal=2)
#assert_almost_equal(res.llf, result_gretl_g1['llf'][1], decimal=7) #not in gretl
#assert_almost_equal(res.rsquared, result_gretl_g1['rsquared'][1], decimal=7) #FAIL
#assert_almost_equal(res.rsquared_adj, result_gretl_g1['rsquared_adj'][1], decimal=7) #FAIL
assert_almost_equal(np.sqrt(res.mse_resid), result_gretl_g1['mse_resid_sqrt'][1], decimal=5)
assert_almost_equal(res.fvalue, result_gretl_g1['fvalue'][1], decimal=0)
assert_almost_equal(res.f_pvalue, result_gretl_g1['f_pvalue'][1], decimal=6)
#assert_almost_equal(res.durbin_watson, result_gretl_g1['dw'][1], decimal=7) #TODO
c = oi.reset_ramsey(res, degree=2)
compare_ftest(c, reset_2, decimal=(2,4))
c = oi.reset_ramsey(res, degree=3)
compare_ftest(c, reset_2_3, decimal=(2,4))
#arch
#sm_arch = smsdia.acorr_lm(res.wresid**2, maxlag=4, autolag=None)
sm_arch = smsdia.het_arch(res.wresid, maxlag=4)
assert_almost_equal(sm_arch[0], arch_4[0], decimal=1)
assert_almost_equal(sm_arch[1], arch_4[1], decimal=2)
'''
Performing iterative calculation of rho...
ITER RHO ESS
1 -0.10734 22530.9
2 -0.10814 22530.9
Model 4: Cochrane-Orcutt, using observations 1959:3-2009:3 (T = 201)
Dependent variable: ds_l_realinv
rho = -0.108136
coefficient std. error t-ratio p-value
-------------------------------------------------------------
const -9.50990 0.990456 -9.602 3.65e-018 ***
ds_l_realgdp 4.37040 0.208146 21.00 2.93e-052 ***
realint_1 -0.579253 0.268009 -2.161 0.0319 **
Statistics based on the rho-differenced data:
Mean dependent var 3.113973 S.D. dependent var 18.67447
Sum squared resid 22530.90 S.E. of regression 10.66735
R-squared 0.676973 Adjusted R-squared 0.673710
F(2, 198) 221.0475 P-value(F) 3.56e-51
rho -0.003481 Durbin-Watson 1.993858
'''
'''
RESET test for specification (squares and cubes)
Test statistic: F = 5.219019,
with p-value = P(F(2,197) > 5.21902) = 0.00619
RESET test for specification (squares only)
Test statistic: F = 7.268492,
with p-value = P(F(1,198) > 7.26849) = 0.00762
RESET test for specification (cubes only)
Test statistic: F = 5.248951,
with p-value = P(F(1,198) > 5.24895) = 0.023:
'''
'''
Test for ARCH of order 4
coefficient std. error t-ratio p-value
--------------------------------------------------------
alpha(0) 97.0386 20.3234 4.775 3.56e-06 ***
alpha(1) 0.176114 0.0714698 2.464 0.0146 **
alpha(2) -0.0488339 0.0724981 -0.6736 0.5014
alpha(3) -0.0705413 0.0737058 -0.9571 0.3397
alpha(4) 0.0384531 0.0725763 0.5298 0.5968
Null hypothesis: no ARCH effect is present
Test statistic: LM = 7.30776
with p-value = P(Chi-square(4) > 7.30776) = 0.120491:
'''
'''
Variance Inflation Factors
Minimum possible value = 1.0
Values > 10.0 may indicate a collinearity problem
ds_l_realgdp 1.002
realint_1 1.002
VIF(j) = 1/(1 - R(j)^2), where R(j) is the multiple correlation coefficient
between variable j and the other independent variables
Properties of matrix X'X:
1-norm = 6862.0664
Determinant = 1.0296049e+009
Reciprocal condition number = 0.013819244
'''
'''
Test for ARCH of order 4 -
Null hypothesis: no ARCH effect is present
Test statistic: LM = 7.30776
with p-value = P(Chi-square(4) > 7.30776) = 0.120491
Test of common factor restriction -
Null hypothesis: restriction is acceptable
Test statistic: F(2, 195) = 0.426391
with p-value = P(F(2, 195) > 0.426391) = 0.653468
Test for normality of residual -
Null hypothesis: error is normally distributed
Test statistic: Chi-square(2) = 20.2792
with p-value = 3.94837e-005:
'''
#no idea what this is
'''
Augmented regression for common factor test
OLS, using observations 1959:3-2009:3 (T = 201)
Dependent variable: ds_l_realinv
coefficient std. error t-ratio p-value
---------------------------------------------------------------
const -10.9481 1.35807 -8.062 7.44e-014 ***
ds_l_realgdp 4.28893 0.229459 18.69 2.40e-045 ***
realint_1 -0.662644 0.334872 -1.979 0.0492 **
ds_l_realinv_1 -0.108892 0.0715042 -1.523 0.1294
ds_l_realgdp_1 0.660443 0.390372 1.692 0.0923 *
realint_2 0.0769695 0.341527 0.2254 0.8219
Sum of squared residuals = 22432.8
Test of common factor restriction
Test statistic: F(2, 195) = 0.426391, with p-value = 0.653468
'''
################ with OLS, HAC errors
#Model 5: OLS, using observations 1959:2-2009:3 (T = 202)
#Dependent variable: ds_l_realinv
#HAC standard errors, bandwidth 4 (Bartlett kernel)
#coefficient std. error t-ratio p-value 95% CONFIDENCE INTERVAL
#for confidence interval t(199, 0.025) = 1.972
partable = np.array([
[-9.48167, 1.17709, -8.055, 7.17e-014, -11.8029, -7.16049], # ***
[4.37422, 0.328787, 13.30, 2.62e-029, 3.72587, 5.02258], #***
[-0.613997, 0.293619, -2.091, 0.0378, -1.19300, -0.0349939]]) # **
result_gretl_g1 = dict(
endog_mean = ("Mean dependent var", 3.257395),
endog_std = ("S.D. dependent var", 18.73915),
ssr = ("Sum squared resid", 22799.68),
mse_resid_sqrt = ("S.E. of regression", 10.70380),
rsquared = ("R-squared", 0.676978),
rsquared_adj = ("Adjusted R-squared", 0.673731),
fvalue = ("F(2, 199)", 90.79971),
f_pvalue = ("P-value(F)", 9.53e-29),
llf = ("Log-likelihood", -763.9752),
aic = ("Akaike criterion", 1533.950),
bic = ("Schwarz criterion", 1543.875),
hqic = ("Hannan-Quinn", 1537.966),
resid_acf1 = ("rho", -0.107341),
dw = ("Durbin-Watson", 2.213805))
linear_logs = [1.68351, 0.430953, 2, "chi2"]
#for logs: dropping 70 nan or incomplete observations, T=133
#(res_ols.model.exog <=0).any(1).sum() = 69 ?not 70
linear_squares = [7.52477, 0.0232283, 2, "chi2"]
#Autocorrelation, Breusch-Godfrey test for autocorrelation up to order 4
lm_acorr4 = [1.17928, 0.321197, 4, 195, "F"]
lm2_acorr4 = [4.771043, 0.312, 4, "chi2"]
acorr_ljungbox4 = [5.23587, 0.264, 4, "chi2"]
#break
cusum_Harvey_Collier = [0.494432, 0.621549, 198, "t"] #stats.t.sf(0.494432, 198)*2
#see cusum results in files
break_qlr = [3.01985, 0.1, 3, 196, "maxF"] #TODO check this, max at 2001:4
break_chow = [13.1897, 0.00424384, 3, "chi2"] # break at 1984:1
arch_4 = [3.43473, 0.487871, 4, "chi2"]
normality = [23.962, 0.00001, 2, "chi2"]
het_white = [33.503723, 0.000003, 5, "chi2"]
het_breusch_pagan = [1.302014, 0.521520, 2, "chi2"] #TODO: not available
het_breusch_pagan_konker = [0.709924, 0.701200, 2, "chi2"]
reset_2_3 = [5.219019, 0.00619, 2, 197, "f"]
reset_2 = [7.268492, 0.00762, 1, 198, "f"]
reset_3 = [5.248951, 0.023, 1, 198, "f"] #not available
cond_1norm = 5984.0525
determinant = 7.1087467e+008
reciprocal_condition_number = 0.013826504
vif = [1.001, 1.001]
names = 'date residual leverage influence DFFITS'.split()
cur_dir = os.path.abspath(os.path.dirname(__file__))
fpath = os.path.join(cur_dir, 'results/leverage_influence_ols_nostars.txt')
lev = np.genfromtxt(fpath, skip_header=3, skip_footer=1,
converters={0:lambda s: s})
#either numpy 1.6 or python 3.2 changed behavior
if np.isnan(lev[-1]['f1']):
lev = np.genfromtxt(fpath, skip_header=3, skip_footer=2,
converters={0:lambda s: s})
lev.dtype.names = names
res = res_ols #for easier copying
cov_hac = sw.cov_hac_simple(res, nlags=4, use_correction=False)
bse_hac = sw.se_cov(cov_hac)
assert_almost_equal(res.params, partable[:,0], 5)
assert_almost_equal(bse_hac, partable[:,1], 5)
#TODO
assert_almost_equal(res.ssr, result_gretl_g1['ssr'][1], decimal=2)
assert_almost_equal(res.llf, result_gretl_g1['llf'][1], decimal=4) #not in gretl
assert_almost_equal(res.rsquared, result_gretl_g1['rsquared'][1], decimal=6) #FAIL
assert_almost_equal(res.rsquared_adj, result_gretl_g1['rsquared_adj'][1], decimal=6) #FAIL
assert_almost_equal(np.sqrt(res.mse_resid), result_gretl_g1['mse_resid_sqrt'][1], decimal=5)
#f-value is based on cov_hac I guess
#res2 = res.get_robustcov_results(cov_type='HC1')
# TODO: fvalue differs from Gretl, trying any of the HCx
#assert_almost_equal(res2.fvalue, result_gretl_g1['fvalue'][1], decimal=0) #FAIL
#assert_approx_equal(res.f_pvalue, result_gretl_g1['f_pvalue'][1], significant=1) #FAIL
#assert_almost_equal(res.durbin_watson, result_gretl_g1['dw'][1], decimal=7) #TODO
c = oi.reset_ramsey(res, degree=2)
compare_ftest(c, reset_2, decimal=(6,5))
c = oi.reset_ramsey(res, degree=3)
compare_ftest(c, reset_2_3, decimal=(6,5))
linear_sq = smsdia.linear_lm(res.resid, res.model.exog)
assert_almost_equal(linear_sq[0], linear_squares[0], decimal=6)
assert_almost_equal(linear_sq[1], linear_squares[1], decimal=7)
hbpk = smsdia.het_breuschpagan(res.resid, res.model.exog)
assert_almost_equal(hbpk[0], het_breusch_pagan_konker[0], decimal=6)
assert_almost_equal(hbpk[1], het_breusch_pagan_konker[1], decimal=6)
hw = smsdia.het_white(res.resid, res.model.exog)
assert_almost_equal(hw[:2], het_white[:2], 6)
#arch
#sm_arch = smsdia.acorr_lm(res.resid**2, maxlag=4, autolag=None)
sm_arch = smsdia.het_arch(res.resid, maxlag=4)
assert_almost_equal(sm_arch[0], arch_4[0], decimal=5)
assert_almost_equal(sm_arch[1], arch_4[1], decimal=6)
vif2 = [oi.variance_inflation_factor(res.model.exog, k) for k in [1,2]]
infl = oi.OLSInfluence(res_ols)
#print np.max(np.abs(lev['DFFITS'] - infl.dffits[0]))
#print np.max(np.abs(lev['leverage'] - infl.hat_matrix_diag))
#print np.max(np.abs(lev['influence'] - infl.influence)) #just added this based on Gretl
#just rough test, low decimal in Gretl output,
assert_almost_equal(lev['residual'], res.resid, decimal=3)
assert_almost_equal(lev['DFFITS'], infl.dffits[0], decimal=3)
assert_almost_equal(lev['leverage'], infl.hat_matrix_diag, decimal=3)
assert_almost_equal(lev['influence'], infl.influence, decimal=4)
def test_GLSARlag():
#test that results for lag>1 is close to lag=1, and smaller ssr
from statsmodels.datasets import macrodata
d2 = macrodata.load_pandas().data
g_gdp = 400*np.diff(np.log(d2['realgdp'].values))
g_inv = 400*np.diff(np.log(d2['realinv'].values))
exogg = add_constant(np.c_[g_gdp, d2['realint'][:-1].values], prepend=False)
mod1 = GLSAR(g_inv, exogg, 1)
res1 = mod1.iterative_fit(5)
mod4 = GLSAR(g_inv, exogg, 4)
res4 = mod4.iterative_fit(10)
assert_array_less(np.abs(res1.params / res4.params - 1), 0.03)
assert_array_less(res4.ssr, res1.ssr)
assert_array_less(np.abs(res4.bse / res1.bse) - 1, 0.015)
assert_array_less(np.abs((res4.fittedvalues / res1.fittedvalues - 1).mean()),
0.015)
assert_equal(len(mod4.rho), 4)
if __name__ == '__main__':
t = TestGLSARGretl()
t.test_all()
'''
Model 5: OLS, using observations 1959:2-2009:3 (T = 202)
Dependent variable: ds_l_realinv
HAC standard errors, bandwidth 4 (Bartlett kernel)
coefficient std. error t-ratio p-value
-------------------------------------------------------------
const -9.48167 1.17709 -8.055 7.17e-014 ***
ds_l_realgdp 4.37422 0.328787 13.30 2.62e-029 ***
realint_1 -0.613997 0.293619 -2.091 0.0378 **
Mean dependent var 3.257395 S.D. dependent var 18.73915
Sum squared resid 22799.68 S.E. of regression 10.70380
R-squared 0.676978 Adjusted R-squared 0.673731
F(2, 199) 90.79971 P-value(F) 9.53e-29
Log-likelihood -763.9752 Akaike criterion 1533.950
Schwarz criterion 1543.875 Hannan-Quinn 1537.966
rho -0.107341 Durbin-Watson 2.213805
QLR test for structural break -
Null hypothesis: no structural break
Test statistic: max F(3, 196) = 3.01985 at observation 2001:4
(10 percent critical value = 4.09)
Non-linearity test (logs) -
Null hypothesis: relationship is linear
Test statistic: LM = 1.68351
with p-value = P(Chi-square(2) > 1.68351) = 0.430953
Non-linearity test (squares) -
Null hypothesis: relationship is linear
Test statistic: LM = 7.52477
with p-value = P(Chi-square(2) > 7.52477) = 0.0232283
LM test for autocorrelation up to order 4 -
Null hypothesis: no autocorrelation
Test statistic: LMF = 1.17928
with p-value = P(F(4,195) > 1.17928) = 0.321197
CUSUM test for parameter stability -
Null hypothesis: no change in parameters
Test statistic: Harvey-Collier t(198) = 0.494432
with p-value = P(t(198) > 0.494432) = 0.621549
Chow test for structural break at observation 1984:1 -
Null hypothesis: no structural break
Asymptotic test statistic: Chi-square(3) = 13.1897
with p-value = 0.00424384
Test for ARCH of order 4 -
Null hypothesis: no ARCH effect is present
Test statistic: LM = 3.43473
with p-value = P(Chi-square(4) > 3.43473) = 0.487871:
#ANOVA
Analysis of Variance:
Sum of squares df Mean square
Regression 47782.7 2 23891.3
Residual 22799.7 199 114.571
Total 70582.3 201 351.156
R^2 = 47782.7 / 70582.3 = 0.676978
F(2, 199) = 23891.3 / 114.571 = 208.528 [p-value 1.47e-049]
#LM-test autocorrelation
Breusch-Godfrey test for autocorrelation up to order 4
OLS, using observations 1959:2-2009:3 (T = 202)
Dependent variable: uhat
coefficient std. error t-ratio p-value
------------------------------------------------------------
const 0.0640964 1.06719 0.06006 0.9522
ds_l_realgdp -0.0456010 0.217377 -0.2098 0.8341
realint_1 0.0511769 0.293136 0.1746 0.8616
uhat_1 -0.104707 0.0719948 -1.454 0.1475
uhat_2 -0.00898483 0.0742817 -0.1210 0.9039
uhat_3 0.0837332 0.0735015 1.139 0.2560
uhat_4 -0.0636242 0.0737363 -0.8629 0.3893
Unadjusted R-squared = 0.023619
Test statistic: LMF = 1.179281,
with p-value = P(F(4,195) > 1.17928) = 0.321
Alternative statistic: TR^2 = 4.771043,
with p-value = P(Chi-square(4) > 4.77104) = 0.312
Ljung-Box Q' = 5.23587,
with p-value = P(Chi-square(4) > 5.23587) = 0.264:
RESET test for specification (squares and cubes)
Test statistic: F = 5.219019,
with p-value = P(F(2,197) > 5.21902) = 0.00619
RESET test for specification (squares only)
Test statistic: F = 7.268492,
with p-value = P(F(1,198) > 7.26849) = 0.00762
RESET test for specification (cubes only)
Test statistic: F = 5.248951,
with p-value = P(F(1,198) > 5.24895) = 0.023
#heteroscedasticity White
White's test for heteroskedasticity
OLS, using observations 1959:2-2009:3 (T = 202)
Dependent variable: uhat^2
coefficient std. error t-ratio p-value
-------------------------------------------------------------
const 104.920 21.5848 4.861 2.39e-06 ***
ds_l_realgdp -29.7040 6.24983 -4.753 3.88e-06 ***
realint_1 -6.93102 6.95607 -0.9964 0.3203
sq_ds_l_realg 4.12054 0.684920 6.016 8.62e-09 ***
X2_X3 2.89685 1.38571 2.091 0.0379 **
sq_realint_1 0.662135 1.10919 0.5970 0.5512
Unadjusted R-squared = 0.165860
Test statistic: TR^2 = 33.503723,
with p-value = P(Chi-square(5) > 33.503723) = 0.000003:
#heteroscedasticity Breusch-Pagan (original)
Breusch-Pagan test for heteroskedasticity
OLS, using observations 1959:2-2009:3 (T = 202)
Dependent variable: scaled uhat^2
coefficient std. error t-ratio p-value
-------------------------------------------------------------
const 1.09468 0.192281 5.693 4.43e-08 ***
ds_l_realgdp -0.0323119 0.0386353 -0.8363 0.4040
realint_1 0.00410778 0.0512274 0.08019 0.9362
Explained sum of squares = 2.60403
Test statistic: LM = 1.302014,
with p-value = P(Chi-square(2) > 1.302014) = 0.521520
#heteroscedasticity Breusch-Pagan Koenker
Breusch-Pagan test for heteroskedasticity
OLS, using observations 1959:2-2009:3 (T = 202)
Dependent variable: scaled uhat^2 (Koenker robust variant)
coefficient std. error t-ratio p-value
------------------------------------------------------------
const 10.6870 21.7027 0.4924 0.6230
ds_l_realgdp -3.64704 4.36075 -0.8363 0.4040
realint_1 0.463643 5.78202 0.08019 0.9362
Explained sum of squares = 33174.2
Test statistic: LM = 0.709924,
with p-value = P(Chi-square(2) > 0.709924) = 0.701200
########## forecast
#forecast mean y
For 95% confidence intervals, t(199, 0.025) = 1.972
Obs ds_l_realinv prediction std. error 95% interval
2008:3 -7.134492 -17.177905 2.946312 -22.987904 - -11.367905
2008:4 -27.665860 -36.294434 3.036851 -42.282972 - -30.305896
2009:1 -70.239280 -44.018178 4.007017 -51.919841 - -36.116516
2009:2 -27.024588 -12.284842 1.427414 -15.099640 - -9.470044
2009:3 8.078897 4.483669 1.315876 1.888819 - 7.078520
Forecast evaluation statistics
Mean Error -3.7387
Mean Squared Error 218.61
Root Mean Squared Error 14.785
Mean Absolute Error 12.646
Mean Percentage Error -7.1173
Mean Absolute Percentage Error -43.867
Theil's U 0.4365
Bias proportion, UM 0.06394
Regression proportion, UR 0.13557
Disturbance proportion, UD 0.80049
#forecast actual y
For 95% confidence intervals, t(199, 0.025) = 1.972
Obs ds_l_realinv prediction std. error 95% interval
2008:3 -7.134492 -17.177905 11.101892 -39.070353 - 4.714544
2008:4 -27.665860 -36.294434 11.126262 -58.234939 - -14.353928
2009:1 -70.239280 -44.018178 11.429236 -66.556135 - -21.480222
2009:2 -27.024588 -12.284842 10.798554 -33.579120 - 9.009436
2009:3 8.078897 4.483669 10.784377 -16.782652 - 25.749991
Forecast evaluation statistics
Mean Error -3.7387
Mean Squared Error 218.61
Root Mean Squared Error 14.785
Mean Absolute Error 12.646
Mean Percentage Error -7.1173
Mean Absolute Percentage Error -43.867
Theil's U 0.4365
Bias proportion, UM 0.06394
Regression proportion, UR 0.13557
Disturbance proportion, UD 0.80049
'''